[petalas]

Result

Finite certificate on two pinned witnesses (scoped negative result); no new κ. On both sharp witnesses of the A8 simplex nine-term decoder on P120 (#1248), the separate-term COPY bill stays at 72 against arbitrary nonlinear zero-error messages when every reader knows all inputs orthogonal to it. This answers the second Open next question of #5299 (research/a7-a1-nonlinear-copy-floor). That question noted that #5299's Lemma 1 transfers to A8 at once and that its Lemma 2 needs its own witness search. This PR supplies the search.

Setting	Bound	Scope
no side information	72	every complex shift (#5299 Lemma 1 with #1248's theorem)
orthogonal side information	72	both #1248 sharp witnesses (new finite certificate)
same, independence bound only	64	both witnesses (exact maximum, control)

Per term, verify.py finds and checks an invertible fooling block of size rho_k, plus an induced Petersen with rho_k - 5 isolated lines inside the term's demand lines X_k ∩ Y_k, or a pairwise non-orthogonal set of size rho_k:

• dropped witness 0,9^8: 8 × (Petersen + 4) on 50-line demand graphs;
• nine-active witness 1,8,9^7: one single line, Petersen + 3 on 49 lines, and 7 × (Petersen + 4).

The largest pairwise non-orthogonal set is computed exactly by branch and bound. It has size 8 in every rank-9 term and 7 in the rank-8 term, so the independence bound alone certifies only 64. The Petersen step supplies the missing unit in all 16 high-rank terms.

Evidence and reproduction

Standard library only. From the repository root:

~/git/integer-mult-bounds/.venv/bin/python3.12 -I -B research/a8-simplex-side-information-floor/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -I -B research/a8-simplex-side-information-floor/test_verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/a8-simplex-side-information-floor
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/a8-simplex-side-information-floor --check-source-history

(The interpreter is CPython 3.12; any python3.12 works.) Local results at e374f75c9bfb625695a14c4e4daa621adb8008bd:

PASS_A8_SIMPLEX_SIDE_INFORMATION_FLOOR roots=120 witnesses=2 no_side_info=72 orthogonal_side_info=72 independence_only=64 petersen_terms=16 certificate=e1b19301ed2e402d no_new_kappa
PASS_A8_SIMPLEX_SIDE_INFORMATION_CONTROLS rejected=12
PASS_REGISTERED_RESEARCH_PACKAGE research/a8-simplex-side-information-floor outcome=negative-result
PASS_RESEARCH_COMMIT_BYTES ... "source_history_commits": ["19fdb086ea96af9a9f87936a2b888e10af4ed810"]

Each registered check takes about 0.1 s locally; the full package run takes about 0.5 s.

verify.py checks the SHA256 of the pinned #1248 CERTIFICATE.json and rebuilds the 120-line A8 model from coordinates. It does not import predecessor code. It checks all 14,400 entries of D T^t = 2G for both witnesses and recomputes both invoices from the pairing matrix, requiring them to equal the pinned ones. It then rebuilds the certificates and requires byte equality with the stored RESULT.json, and re-checks the stored certificate as given. The twelve rejecting controls cover:

• a singular fooling block and a fooling line outside the term;
• swapped Petersen labels, an orthogonal "isolated" line, and a duplicated line;
• Petersen + 3 against rank 9, a non-independent set, and an independence-only claim;
• an invoice mismatch, a broken identity, a tampered stored certificate, and a drifted input.

Limitations and unrun obligations

• Only Research: sharp A8 simplex nine-term decoder copy floor 72 (no new kappa) #1248's two pinned sharp witnesses are certified under side information. The every-shift side-information question remains open.
• The two rate lemmas are Research: nonlinear messages keep the A7+A1 nine-term COPY floor at 72 (no new kappa) #5299's written lemmas, used as premises: the fooling bound, and the Petersen + isolated-receiver induction. The Petersen broadcast rate 5 is Blasiak–Kleinberg–Lubetzky's published Fact 5.6(1). None of these is re-proved here.
• No independent human proof review or formal proof-kernel check is claimed. No supplier, physical circuit, frontier change or κ is claimed.

Open next questions

1. Does the orthogonal-side-information floor stay at 72 for every complex row shift of the A8 simplex decoder? A first step is to list support-reducing shifts and find, per term, the largest demand set with neither a Petersen + (rank − 5) witness nor a rank-sized non-orthogonal set.
2. Is the Petersen step forced, or would another induced subgraph whose broadcast rate exceeds its independence number close the one-unit gap in each rank-9 term? A first step is to compare LP or fractional index-coding bounds with the independence number on one 50-line demand graph.

Related open work

• No open PR studies the A8 simplex decoder or the side-information COPY floor (checked open PRs and merged searches for "A8 simplex", "side information" and "nonlinear copy floor").
• Prove the F′ certificate ceiling 360/17 on P120, down from 24 (no new κ) #5956 (F′ certificate ceiling on P120) and Prove P112 quadratic extension threshold is exactly 7 (no new κ) #5957 (P112 quadratic-extension threshold) study other P120/P112 decoder bounds. They are different quantities, and no number is shared.
• Bound E6 and E7 F2 shadow minranks by 8 and 11 (no new κ) #5959 (E6/E7 F2 shadow minranks) studies characteristic-two minranks of other root frames. It is a different quantity, and no number is shared.

Credit

Mark (Matarener)'s #5299 poses the question and supplies both lemmas and the certificate format. Philipp / mightyOil's #1248 supplies the A8 model, the every-shift floor 72 and both witnesses. joaqcoo's #1224 supplies the one-sided normal form. Width pricing follows #888 and #785/#814.

Claim: #1166 (comment)

CI notes

All checks passed at e374f75c9bfb625695a14c4e4daa621adb8008bd: plan and replay (research/a8-simplex-side-information-floor), run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40533181921.